The Bell System Technical Journal : devoted to the Scientific and Engineering aspects of Electrical Communication, Vol. 22, No. 3

P . 2S/ i / 3

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION

I . %*, Vr*' B

| fjp- CjiL I

\ r /

Effect of Feedback on Impedance. . . R. B. Blackman 269 Design of Two-Terminal Balancing Networks

—K . G. Van Wynen 278 Use of X-Rays for Determining the Orientation of Quartz

Crystals

— W. L. Bond and E. J. Armstrong 293 Raw Quartz, Its Imperfections and Inspection (Chapter IV)

— G. W. Willard 3 3 8 The New Statistical Mechanics . . . Karl K . Darrow 362 Review of S. A. Schelkunoff’s Book

Electromagnetic W a v e s P. Le Corbeiller 3 93

Abstracts of Technical Articles by Bell System Authors . 397 Contributors to this I s s u e ... 402

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T h e Bell System Technical Journal

Vol. X X I I October

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¹⁹⁴³

E ffect of F eed b a ck on Im p ed an ce

By R. B. BLACKMAN

T

H E impedance of a network is defined as the complex ratio of the alter­

nating potential difference m aintained across its terminals by an ex­

ternal source of electromotive force, to the resulting current flowing into these term inals. If the network contains active elements such as vacuum tubes, the resulting current (or potential difference if the input current is taken as the independent variable) m ay be due in p a rt to the excitation of the active elements. The definition of impedance does not discriminate between the p a rt of the current (or potential difference) due directly to the external source of electromotive force and the p a rt due to the excitation of the active elements by the external source. Hence the impedance will in general depend upon the degree of activity of the active elements.

These observations were made early in the development of feedback amplifiers by H. S. Black1 who made two im portant uses of the effect of feedback on impedance. In the first place it afforded a method of m easur­

ing feedback which has some advantages over the m ethod which involves opening the feedback loop, providing proper term inations for it and meas­

uring the transm ission around it. In the second place the effect of feedback on impedance was used to control the impedances presented by a feedback amplifier to the external circuits connected to it.

Relations between impedance and feedback were derived by Black and others for a num ber of specific feedback amplifier configurations. In some cases these relations turned out to be very simple. For the most p art, how­

ever, these relations wete so complicated th a t they defied reduction to a common form .2 The difficulty seems to have been due, in p a rt a t least, to the a ttem p t to form ulate the relationship, in each case, in term s of the nor­

mal feedback of the amplifier. In some cases the difficulty seems to have been due p artly also to the valid, but, as it turns out, irrelevant observation th a t the feedback is affected by the impedance of the measuring circuit as

1 H. S. Black, “ Stabilized Feedback Amplifiers” , B .S .T .J ., January, 1934.

2 Shortly after the general relationship between feedback and impedance was derived, it was independently established by H. W. Bode and J. M. W est by exam ination of a variety of feedback amplifier designs. T he generality of the relationship was also in­

dependently proved for amplifiers w ith a single feedback p a th by J. G. Kreer and by C.

H. Elmendorf.

269

well as by the rem oval of an y impedance elements or circuits which are norm ally connected to the amplifier.

These difficulties are avoided by the m ethod of derivation adopted in this paper. Illustrative examples are then given of some of the uses to which the general relationship between feedback and impedance m ay be put.

De r i v a t i o n

T he derivation of the general relationship between feedback and im­

pedance will be m ade here w ith reference to the diagram shown in Fig. 1.

One of the vacuum tubes in the network, nam ely th a t one to which the feedback is to be referred, is shown explicitly a t the top of the box in the diagram . T he grid lead to this tube is broken a t term inals 2, 2'. In prac­

tice, the break in the grid lead would leave the grid still coupled to some

degree to the other electrodes of the tube through parasitic interelectrode adm ittance. F or analytical purposes, however, it m ay be assum ed th a t the parasitic adm ittances between the grid and the other electrodes of the tubes are connected not directly to the grid w ithin the tube b u t to some point farther out along the grid lead. U nder this assum ption the break in the grid lead not only removes the feedback to the tube completely, b u t also leaves the parasitic adm ittances connected in the netw ork in such a way th a t their contribution to the feedback is implicitly taken into account.

Furtherm ore, the impedance looking into the grid of the tube is now infinite so th a t if a voltage is applied to the grid no current will be draw n from the source of the voltage.

A t the left-hand side of the box in the diagram , term inals 1, 1' are brought out. These are the term inals to which the impedance is to be referred. In

E F F E C T OF F E E D B A C K ON I M P E D A N C E 271

the norm al condition of the network these terminals m ay be connected through a h external impedance branch. This is the case, for example, when term inals 1, 1' are the input term inals of a feedback amplifier whose input impedance is under investigation. However, this external impedance may also be zero or infinite according as term inals 1, V are “m esh-term inals”

obtained by breaking open a mesh of the network, or “junction-term inals”

obtained by bringing out two junctions of the network.

I t is assumed th a t the network, including all of the vacuum tubes, is a linear system in which, therefore, the Superposition Principle holds. Hence, if an e.m.f. E i is applied in series with term inals 1, 1' and a second e.m.f.

E i is applied between the grid and the cathode of the tube, the potential difference V\ developed across the input term inals 1, 1' and the potential difference V 2 developed between the term inal 2 and the cathode of the tube will be linearly related to E 1 and E 2. If the source of E i has internal im­

pedance the coefficients in these relations will depend upon this impedance.

However, if the input current I i is used as an independent variable in place of the e.m.f. E i the coefficients will not depend upon the impedance of the source of the current h . I t is also convenient to consider the potential difference E2 — V2 developed across the term inals 2, 2' as one of the de­

pendent variables in place of V 2. Therefore, Vi = A h + B E 2 E 2 - V2 = C h + D E i where the coefficients are independent of Z.

From these equations we obtain

(1)

= A D - B C e 2=y 2 D

A

A D - B C

Hence

©

( h )

⁼

\ / l /e2= 0 ( E i - F A

\ E2 ) vi=o ( 5 ç Z » ) . D

\ Ei / i l=o

/ F , \ / F A

This equation expresses the relationship between feedback an d impedance.

To m ake this more ap p are n t th e physical significance of each of the factors in t his equation will be examined an d suitable symbols will be substituted for them .

In equations (1) £2 and 11 were regarded as independent variables. How­

ever, the ratio (-7^ ) implies th a t £2 is adjusted to be equal to V 2.

\ I i / e2=V2

This means th a t £2 is dependent upon I\. The reason for the imposition of this dependence is th a t w ith £2 equal to V2 th e term inals 2, 2' m ay be connected together and the source of £ 2 m ay be rem oved w ithout affecting, in particular, the potential difference V \ across term inals 1, an d the cur­

ren t 11 into these term inals.

Obviously, therefore, the ratio ( ^ ) is the impedance which will be

\ 1 i/ e2=v2

seen a t the term inals 1, V when term inals 2, 2' are connected together and the only source of e.m.f. acting on the netw ork is the external circuit con­

nected to the term inals 1, 1'. This ratio will be symbolized by Z A.

The ratio ( — ) implies th a t no voltage is applied between th e grid

\ /i/k2-o

and the cathode of the tube. However, it is im m aterial w hether or n o t a voltage is applied to the grid of the tube if the am plification of th e tu b e is nullified. Obviously, therefore, this ratio is the impedance which will be seen a t the term inals 1, 1' when term inals 2, 2' are connected together and the amplification of th e tube is nullified. This ratio will be symbolized by ZP.

( V t \ / f a

Finally, th e ratios I y r I a n d I y r ) are readily recognized from V - c ^ / r ^ \ E 2 j Il=o

the definition of feedback to be the feedback to the vacuum tube w ith the term inals 1, 1' connected together in th e first case, an d left open in th e sec­

ond. These ratios will be symbolized by FSh an d F0p respectively.

Hence, equation (2) m ay be w ritten in th e more significant form Z A 1 Fsh

Z~P “ f ^ £ 7 P w

De t e r m i n a t i o n o f Fe e d b a c k

One of the uses to which th e relationship (3) m ay be p u t is in th e deter­

m ination of feedback b y impedance m easurem ent. However, since this relationship involves two feedbacks, only one of which m ay be identified w ith the feedback to be determ ined, one of these feedbacks m ust be known.

In the most common types of feedback amplifiers it is possible to choose

E F F E C T OF F E E D B A C K O N I M P E D A N C E 27 3

term inals 1, 1' so th a t either FSh or FoP is zero. If F0p = 0 and FSh = Fn

where FN is the norm al feedback, then

Fn = 1 - (4)

Zp On the other hand, if Fst, = 0 and F0p = FN then

Fig. 2 shows a feedback amplifier in which the ¿¿-circuit and the / 3 - n e t w o r k

are connected in series a t one end and in parallel a t the other end. A t term inals 1, 1' in this figure the conditions for formula (4) are obviously fulfilled. Hence, if the impedance measurements are made a t these ter-

u - C IR C U IT

Zi

1 r

(5 - N E T W O R K

Fi g. 2— F eedback amplifier w ith series feedback a t one end and sh u n t feedback a t the other end.

minals, the feedback is given by formula (4). On the other hand, a t ter­

minals 2, 2' in Fig. 2 the conditions for formula (5) are obviously fulfilled.

Hence, if the impedance measurements are made a t these term inals, the feedback is given by formula (5).

If the grid-plate parasitic adm ittance of a tube in a feedback amplifier is not negligible it is not possible to open any physical mesh in the amplifier so th a t F0p = 0 for th a t tube. In such a case, therefore, (4) is not ap ­ plicable. However, if the impedance measurements are made between the grid and the cathode of th a t tube the conditions for formula (5) are ob­

viously fulfilled, and the feedback is given by formula (5). Hence, of the two particular forms (4) and (5) of the general relationship (3), only (5) enjoys complete generality in the determ ination of feedback by impedance measurements.

Fe e d b a c k d u r i n g Im p e d a n c e Me a s u r e m e n t s

W hile the feedback com puted from impedance m easurem ents by formula (4) or (5) is the norm al feedback, the feedback during the impedance meas­

urem ents m ay be quite different, due to the impedance of the impedance m easuring circuit. Referring to Fig. 1 we see th a t the feedback during m easurem ent is by definition

where Z is the impedance of the impedance m easuring circuit. B y equa­

tions (1) this is easily reduced to

I t is clear therefore th a t even if FN satisfies N y q u ist’s S tability Criterion, Z m ay be of such a character th a t Fz violates th a t criterion. In th a t case it will be impossible to m ake the impedance m easurements.

Contrariwise, if FN violates N y q u ist’s Stability Criterion, it is possible to choose Z so th a t F z satisfies th a t criterion and make it possible to m eas­

ure the impedance. S ubstituting (4) into (7) we find th a t a sufficient b u t not necessary condition in order th a t \FZ\< 1 is th a t

Similar observations m ay be made w ith respect to (8) as were m ade with respect to (7). S ubstituting (5) into (8) we find th a t a sufficient b u t not necessary condition in order th a t |F Z| < 1 is th a t

(

6

)

(7)

\ Z \ > \ Z A \ + 2 \ Z P \ U nder the conditions to which form ula (5) applies

(⁸)

I ¿ I \ ¿ A \ \ ¿p [

Fe e d b a c k Co n t r o l o f Im p e d a n c e

T he application of the relationship (3) to the feedback control of im­

pedance m ay be illustrated by a few concrete examples.

E F F E C T OF F E E D B A C K O N I M P E D A N C E 275

L et us assume th a t we are interested in the impedance faced by the line impedance Z\ in Fig. 2. If the terminals 1, 1' in Fig. 3 are left open the feedback is obviously zero. L et the feedback when the term inals are shorted together be denoted by F®. If the impedances of the ¿¿-circuit and the /3-network are denoted by Z^ and Zp, respectively, then

Za = Z P( 1 - F sh) (9)

where

Z P = Z„ + z $

This shows the now well-known fact th a t series feedback m ay be used to magnify impedance.

Fig. 3— Im pedance faced by the line a t th e series feedback end of a feedback amplifier.

However, it should be noted th a t the feedback F Sh involved in (9) is not now equal to the norm al feedback F N as it was when the term inals 1 ,1 ' were taken as in Fig. 2. The relation between F N and F Sh may be obtained from (6) by identifying F N with F z , and Z i with Z. Hence

Fn = (10)

1+^

From (9) and (10) it follows th a t even with a very modest am ount of normal feedback the magnification of the impedance m ay be very large. For example, if Z P = 1000 ohms, Z \ = 1 megohm and F Sh = — 1000, then Z A is b etter th an 1000 times as large as Z P although F,v is not quite unity in magnitude.

Similarly, the impedance faced by the line impedance Z 2 in Fig. 2, as shown in Fig. 4, is

Z A =

1 - F,O p (1 1 )

where

z , + Zß

This shows the now well-known fact th a t shunt feedback m ay be used to reduce impedance.

T he relation between the norm al feedback F.v and the feedback F 0p in­

volved in (11) is, by (6)

Fn = Fop

‘ + 1

(^{1 2})

From (11) and (12) it follows th a t even w ith a very modest am ount of nor­

m al feedback the reduction in impedance m ay be very large. F or example,

E F F E C T OF F E E D B A C K ON I M P E D A N C E 277

if Z;> = 100,000 ohms, Z 2 = 100 ohms and FnP = —1000, then Z A is less than 100 ohms although FN is not quite unity in magnitude.

T he two examples given above illustrate the use of feedback to magnify or to reduce the impedance of a network. This impedance, however, will be correspondingly sensitive to changes in the characteristics of the vacuum lubes. A third example of the use of the relationship (3) will show th a t feedback m ay also be used to make the impedance of a network less sensitive to changes in the characteristics of the vacuum tubes.

In the case of the bridge-type feedback network shown in Fig. 5 we have, with respect to the term inals 1, 1'

ZP = R ( \ + Q)

2R + 3Rf] n Iah A 1 + Q

Fop = A ^1 + -

where A is the feedback designed for the condition R v = R, and (RP

-

^{R) (R}

+

^Rp)

Q

=

Then, by (3)

(R + R P)(2R + Rp) + 2RR?

Hence, if the feedback F0v is very large the effect of bridge unbalance on the impedance presented to the line will be very small. If, for example, the design feedback is 40 db the ou tp u t impedance cannot change more than 1 per cent however severely the bridge might be unbalanced by R v being larger th an R.

The feedback when the line impedance R L is connected may be obtained by identifying Ri, with Z in formula (6). I t is

2R + 3Rp n

1 + R

R + Rl

whence

d log F _ RRr, Q d log Ri, R + Ri, Rl + Zp

T he effect of bridge unbalance is to make the feedback sensitive to changes in I he line impedance Rl.

By K . G. VanW YNEN

T his paper describes a simple graphical m ethod for designing a tw o-term i­

nal netw ork, w hich will sim ulate a given line im pedance to such a degree th a t retu rn losses of the order of 25 db or b e tte r will be readily obtained. T he m ethod is particularly useful in those problem s in w hich a reasonably accurate balancing netw ork is adequate, b u t a high degree of precision is n ot required.

Ge n e r a l

I

T IS the purpose of this paper to describe a graphical m ethod which has been found useful in the design of simple tw o-term inal netw orks to sim ulate the impedance of transm ission lines or equipm ent. T he dis­

cussion which follows is intended to emphasize the sim plicity of the m ethod and the rap id ity with which it m ay be employed to arrive a t a solution; it will also indicate th e analytical background w ithout a t­

tem pting to develop or establish th e rigor of the procedure involved.

A solution can frequently be obtained in a fraction of an hour and it is thought th a t th e graphical analysis will appeal to th e pragm atist and the engineer who has a job to do, b u t very little tim e in which to accomplish his aim, rath er th a n the person interested in th e rigor of the solution.

T he problem which is considered m ay be stated as follows: Design a tw o-term inal netw ork w ith th e m inim um num ber of elements which will give a desired degree of approxim ation to a given impedance function Z ( \ ) , where Z(X) is a fraction whose num erator and denom inator are poly­

nomials in frequency in accordance w ith the custom ary usage in such prob­

lems.

Or i g i n o f Pr o b l e m

This problem has arisen most generally in providing balancing networks which will give satisfactory retu rn losses against various types of telephone facilities. I t is obvious th a t for a given impedance, (r + j x ) , a t a given frequency there are an infinite num ber of netw orks which will satisfy th e given impedance. I t has also been pointed out th a t th e netw ork which sim ulates a given impedance function is n o t unique. H ence there are also a large num ber of netw orks which will satisfy a given impedance function.

In designing netw orks for repeater circuits, it is generally satisfactory 278

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 279

if th e retu rn loss is equal to or greater th a n some specified num ber of db.

This somewhat simplifies our problem and perm its a double infinity of solutions. A m ethod has been given by Brune1 for designing such networks, in which it is pointed out th a t there is no unique solution to the problem of designing a finite tw o-term inal netw ork and also states th a t any netw ork which satisfies the impedance function m ay be considered a satisfactory solution to the problem. I t is thought th a t th e m ethod which is given below will provide a solution which makes maximum use of the num ber of elements employed. T h a t is, it will provide a given retu rn loss w ith the minim um num ber of parts.

T he required degree of approxim ation and th e frequency range to be covered determ ine th e num ber of elements required in this solution. In one simple case which will be discussed below in the first example, the approxi­

m ation between the impedance of a transm ission line and a netw ork designed to sim ulate it is th e approxim ation between the curvature of the impedance function and th e arc of a circle.

Ge n e r a t i n g Fu n c t i o n

The m ethod discussed here differs from th a t outlined by Brune in th a t use is made of known generating functions which are added together in series to approxim ate the to ta l function, similar to the m anner in which sine functions m ay be added to approxim ate other functions. This series type netw ork can readily be converted to the ladder type by well known n et­

work equivalence theorems and the solution will then have the Stieltjes fraction form pointed out by F ry 2 and Cauer.3

T he generating function used here is an impedance consisting of a resist­

ance in parallel w ith a pure reactance or a special case of this. This func­

tion plus a real corresponds to a bilinear transform ation, the properties of which have frequently been discussed elsewhere. This particular con­

figuration, for instance, has been pointed out both by Brune and by Guille- m in3 a t M .I.T . and a discussion of th e bilinear transform ation has been given by C. W. C arter4 of the Bell Telephone Laboratories. The series addition of such generating functions is similar to th e form given in F oster’s reactance theorem except th a t there only pure reactances are dealt with.

The solution can also be worked out w ith adm ittances, b u t will not be dis­

cussed here since th e average engineer is more accustomed to dealing w ith impedances.

In m any problems, particularly those involving dissipative transmission 1 Jour. Math. & Physics, Vol X , 1930-1931.

2 Bull. A m . Math. Soc., 35,1929.

3 Guillemin—Vol. II.

4B.S.r./., July 1925.

lines, the entire impedance function is found in the fourth q u adrant of the complex plane. W hen this is so, th e generating function is reduced to a resistance in parallel w ith a condenser.

Gr a p h i c a l Re p r e s e n t a t i o n o e Fu n c t i o n s

As a first step in utilizing the graphical procedure, it will be advisable to acquire some fam iliarity with the generating function in its general form

(B)

R L C X ^ F L X + R

F=oo

Z(X)= R L X L X + R

( C ) R H W l n

F=o

z(xj- R C X + I

( E )

(°) 0—Wv—O X ^{f- ( F} r _JL X

F = o T O F-0

R R

CŁII<"n

Z ( A ) = R + L X

( F >

R C

o-WVo-o-ll-o

F“0O

Z ( X ) = R + ^

F = 0

CG)

o—^ 0 00 o

F-Fa

zOO=L C X 2 - t R C X + l A C

Fig. 1.—T he im pedance loci, Z ( \ ) , for several netw orks.

and some of its special cases. Plots of various cases are given in Figs. 1(a) through 1(h) together w ith th e netw ork configuration and th e impedance function thereof. Obviously th e sum m ation of th e properly selected gen­

erating functions corresponds to th e addition of th e p a rtia l fractions de­

rived by B rune’s m ethod. For an accurate solution these p artial fractions when combined should approxim ate th e given Z(X).

Figure 1(A) shows th e impedance locus of the parallel R, L, C generating

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 281

function as frequency varies from 0 to °o . A t 0 frequency Z ( \ ) = 0 + jO and a t co Z ( \ ) = 0 — jO. The locus is a circle and crosses the real axis at R and the frequency a t which L and C are anti-resonant. T he special cases will be readily apparent and w ithout further discussion attention will be shifted to Fig. 1-c which is the generating function applied to obtain solu­

tion of the examples listed below, all of which are located in the fourth quadrant.

T he impedance of this function (Fig. 1-c) a t zero cycles is a real and has the value R, and a t infinite frequency its impedance is 0 — jO. The locus traced by this function in the fourth quadrant of the complex plane as / varies from zero to infinity is a semicircle of radius R /2 whose center is at R / 2 on the axis of reals. Obviously, the impedance for any given fre­

quency depends only on C when R has been fixed.

One of the most useful networks for voice frequency work is th a t in which two such functions are added together b u t the second function is the special case in which C = 0. We then have a netw ork which consists of a resist­

ance Ri in series with the parallel combination i?2 and C2, and is represented by th e semicircle just described b u t displaced to the right of the origin by the distance Ri. This form corresponds to a special case of the bilinear transform ation previously mentioned.

As stated earlier a given impedance function can be obtained from a large num ber of networks b u t when th e impedance is to be sim ulated for a limited frequency range, such as the voice band, the selection of the best network is reduced to sorting through a relatively small range of networks to select th a t one which is the best compromise for the given conditions. This then is a restatem ent of the problem : To find the network having the minimum number of circuit elements which will give the desired approximation to a speci­

fied impedance function.

T he other sections of Fig. 1 will be evident upon analysis.

Me t h o d o f So l u t i o n

T he first step to be followed in finding the solution to a given problem is to plot in th e complex plane the locus traced by the given impedance function as the frequency varies over th e range which is to be considered and to m ark the frequency a t those impedances which are essential to the problem. H aving done this, the next step is to draw a semicircle with the center on th e real axis such th a t an arc of the semicircle approximates p art or all of the locus of th e impedance function. In m any cases this semi­

circle is a sufficiently good approxim ation but where it is not, it will be necessary to add other functions. T he examples given below are illustra­

tive of cases requiring three-, four- and five-element networks.

Ex a m p l e 1— 104 Mi l Op e n Wi r e

To dem onstrate th e m ethod we will now consider th e design of a n e t­

work which sim ulates a 104-mil copper open-wire line w ith 12 in. spacing and CS insulators. T he impedance function for this p articular facility is p lotted on Fig. 2. I t is perhaps rath er obvious th a t this locus can readily be a p p r o x im a te d by a semicircle whose center is on the real axis and whose intercept on the real axis is n o t a t the origin. Such a semicircle has been drawn, b u t it is recognized th a t the one shown is not unique, for it would be possible to draw several others which m ight do equally well. However, they

Ri

Fig. 2.— G raphical design of tw o-term inal balancing netw ork for 104-mil. copper open wire.

would in general be fairly close to th a t shown. H aving selected this semi­

circle, which approxim ates the impedance function, it is evident th a t a n et­

work consisting of a resistance, R 1 , in series w ith a parallel R2C\ combina tion will provide a reasonable approxim ation above 200 cycles. The series re­

sistance R i , is, of course, the left-hand intercept of the semicircle and the R axis and the parallel resistance, R 2 , is the diam eter of the semicircle. There rem ains, then, th e problem of determ ining C2 which obviously governs the distribution of frequencies along th e semicircular locus. If C2 is very small, th e 1000 cycle impedance will be near th e right-hand end of th e locus since

R2 is controlling and vice versa. The answer as to w hat value of C should be selected depends on w hat frequency range we are m ost interested in approx­

im ating closely. Suppose in this case th a t we say 1000 cycles is the fre­

quency a t which we wish to have the best degree of approxim ation. C2 will then be determ ined by drawing a vertical axis passing through R i and inscribing a semicircle passing through i?i and the 1000-cycle impedance of th e open-wire line function and having its diam eter on th e vertical axis. The diam eter of this semicircle represents X c and therefore determines the capacity, C2, of the parallel combination.

Carrying out th e procedure ju st described it will be seen by reference to Fig. 2, th a t X c2 = 145 ohms a t 1000 cycles and therefore C2 = 1.1 mf.

T he 3-element netw ork thus determ ined is a resistance of 654 ohms in series w ith th e parallel combination of 1800 ohms and 1.1 mf. B y arb itrary choice the 1000-cycle impedance of th e line and netw ork are in good agree­

m ent. I t is now necessary to determ ine th e netw ork impedance a t other frequencies in order to compare them against th e open-wire line impedance.

As is well known the parallel impedance a t any other frequency is the intersection of th e corresponding X c and i ?2 semicircles. A t 200 cycles X c

= 725 ohms. D rawing a semicircle of diam eter 725 ohms on the vertical axis through 654 ohms the netw ork impedance is located a t th e intersec­

tion of this semicircle and the i ?2 semicircle, i.e., a t 900 — j 620.

T hus th e netw ork impedance locus as a function of frequency m ay be completely determ ined over th e desired frequency range and compared with the given impedance locus of th e open wire.

This m ay be done visually. If corresponding points on the two loci are close together, th e sim ulation will be a good one and vice versa. If it is found th a t th e sim ulation is too good a t one frequency and not good enough a t other frequencies, it will be possible to alter the distribution of frequencies along th e locus by changing C2 or th e locus m ay be shifted by changing i ?2 or both C2 and Ri m ay be changed. No specific rule can be stated for this b u t w ith a little experience considerable dexterity m ay be acquired in this sort of juggling and a locus found which will give an approxim ately con­

sta n t approxim ation over a reasonably wide frequency range. As m ay be seen by referring to Fig. 2, it was found th a t a netw ork consisting of a 654- ohm resistance in series w ith th e parallel combination of 1800 ohms and 1.10 mf. gives a very good sim ulation of a 104 mil copper open wire line over th e voice range. As is obvious from the graphical method, the simula­

tion rapidly deteriorates below 200 cycles due to departure of the network locus from th e impedance locus of the open ware line. If it were necessary to improve this low-frequency simulation, it would be necessary to add further generating functions to the design or compromise a t the higher frequencies.

Since this netw ork was intended for use as a balancing network, it was T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 283

then tested in th e laboratory against th e open-wire impedances and found to give fairly high retu rn losses as listed in Table I. The corresponding retu rn losses were also com puted and tabulated. The impedances are given for both the netw ork and the theoretical line a t typical frequencies over the range from 100 cycles to 20,000 cycles.

T he impedance function for an open-wire line is given by the equation

- ( £ r l ) ‘ \G T C \ / (1) Expanding this function by the binomial theorem and taking the first approxim ation and further letting G = 0, th e impedance function becomes

Ta b l e I

104 M il Cti Open W ire, Dry Weather, 12" Spacing, C S Insulators

F r e q . C y c le s

I m p e d a n c e

R e t u r n L o s s o f N e t ­ w o rk v s L in e - d b

N e t w o r k L in e

R e c t. P o l a r R e c t. P o l a r M e a s u r e d C o m p u te d

100 1360—j878 ' 1620/3279 1101—j883 1410/3878 20.9 21.3

200 904—j623 1 097/34.7 865—j562 1032/ 3 3.0 2 7 .4 2 9 .4

300 3 4.4

500 699—j281 754/ 2TT9 712—j273 7 6 4 / 2 0 37 .8 3 9 .6

1000 665—j 143 68I / 12.2 674— j 144 6 8 9 /1 2 .0 4 2.3 4 3 .6

2000 656— j72 6 6 0 / 6 .3 662— j74 6 6 6 / 6 .4 4 5.2 4 7 .8 5000 654— j28 6 5 4 / 2 .4 658— j'32 6 5 9 / 2 .8 43.1 4 8 .4 10000 654— j l 4 .5 6 5 4 / 1.0 653— jl2 6 5 3 / 1.1 39.5 55.5

20000 654— j'7.2 6 5 4 / 0 .5 652— jlO 6 5 2 / 0 .9 51.2

2 ( x )

⁺

(L C )K 2/R ) X (²)

Applying th e m ethod of Brune, this equation yields a netw ork consisting of 646.4 ohms in series w ith a condenser of 1.09 mf. I t will also be ap p aren t th a t eq. (2) has the same form as th a t of Fig. 1(f), i.e.,

= * + y

and by a 1 to 1 comparison of term s it is evident th a t a

R i a n d

( 3 - a )

( 3 - b )

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 285

Including G, the expression for the first approxim ation of the impedance function m ay be w ritten in the form

Z(X) _{i +} ( - - - V\ 2 L 2 C j + G

^ 2C

(4 )

hollowing B rune’s m ethod or noting th e correspondence with the impedance function given in Fig. 1(c), it is apparent th a t the network is a resistance, Ri , of 646.4 ohms in series w ith a parallel i?2C2 combination. C2 is 1.09 mf as before b u t R> computes as 312,000 ohms which is so large compared to 1.09 mf. th a t th e additional resistance provides negligible improvement over th e previous netw ork for th e voice frequency range.

Obviously then, the analytical m ethod requires a t least a second order approxim ation entailing considerable additional analytical work and com­

p u tatio n which will not be carried out here. This points out the advantages of th e graphical m ethod; namely, it is rapid, requires no special skill, and gives a reasonably accurate answer.

Ex a m p l e 2— Si r a l Fo u r Ca b l e— 1320 Fo o t Sp a c i n g—6 Mi l h e n r y Lo a d in g (SP4-1320-6)

In order to indicate th e procedure when two complete RC regenerating functions are required, another example is given which covers an impedance sim ulation of a SP4-1320-6 line. A plot of this impedance function is shown on Fig. 3, and it is a t once obvious th a t two semicircular generating functions should give a reasonably good approxim ation to the given im­

pedance function.

T he m ethod of selecting these functions m ay be somewhat as follows:

Consider first the sim ulation in th e low-frequency range, i.e., 200 cycles to 500 cycles. For this region a semicircle m ay be selected much as in the first example and th e one chosen yields a netw ork consisting of Ri = 480 ohms in series w ith th e R2C2 parallel combination in which i ?2 = 1460 ohms. C2 was found by choosing an X a t 500 cycles close to th a t of the line and from which C2 was found to be 1.38 mf.

I t is evident th a t to provide high-frequency simulation a condenser m ust be placed in parallel with R x = 480 ohms. Its value is determined by the intersection of the Ri and X Cl semicircles a t 10,000 cycles and C\ is found to be .0161 mf. The construction lines involved in these determ inations are shown as light weight solid lines.

Since there are now two impedance functions to be added in series the locus will depart somewhat from the two semicircles. However, the de­

p artu re will not be great since the effect of Cx is small a t low frequencies,

Fig. 3— G raphical design of tw o-term inal b alan cin g n etw ork for spiral four cable.

Ta b l e I I

Spiral Four Cable— 1320 Foot Spacing— 6 M ilhenry Loading— Full Section Termination

F r e q . C y c le s

I m p e d a n c e R e t u r n L o s s v s

T h e o r e t i c a l L in e -d b

M e a s u r e d R e t u r n L o s s v s A rtif ic ia l*

L in e - d b

N e t w o r k L in e

R e c t . P o l a r R e c t . P o l a r M e a s ­

u r e d C o m ­ p u t e d

100 1046— j713 1 2 6 6 /3 4 3 810—j670 1051/3976 14.8 19.8 21.8 200 701— j'528 8 7 8 /3 7 .0 630—j430 7 6 3 /3 4 .3 22.8 2 2.6 22.9 500 516— j239 5 6 9 /2 4 .9 500— j220 5 4 6 /2 3 .8 33.2 3 3 .0 25.2 1000 488—jl3 9 5 0 7 /1 5 .9 470— jl3 0 4 8 8 /1 5 .5 37.1 33.9 2 6.2 2000 479—jl0 8 4 9 1 /1 2 .7 470—j 100 4 8 1 /1 2 .0 4 1.2 38.1 2 6 .6

3000 2 6.2

5000 456— jl2 5 4 7 3 /1 5 .3 460— j 120 4 7 5 /1 4 .6 39.9 4 3 .4 26.3 7000 430— j 163 4 6 0 /2 0 .8 450— jl5 0 4 7 5 /1 8 .4 3 6.7 3 1 .8 25.1 10000 389— j201 4 3 8 /2 7 .3 420—j200 4 6 5 /2 5 .5 3 2 .4 2 9.3 23.5

12000 2 3 .6

15000 328—j231 4 0 1 /3 5 .1 380—j280 4 7 2 /3 6 .4 21 .8

* 120 sections term in ated in 450 ohms.

and th a t of i ?2 is small a t high frequencies. In this case th e two functions m ay be thought of as virtually independent.

T able I I gives the theoretical impedance of this facility and the com puted

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 287

impedance of the netw ork a t frequencies from 100 cycles to 15,000 cycles and, as m ay be seen by the comparison, a fairly good simulation exists throughout the range. This fact has been verified by making return loss measurements in the laboratory against the theoretical line with the results indicated in the table. R eturn loss m easurem ents have also been made be­

tween the netw ork and an artificial line consisting of 120 sections of this facility term inated in 450 ohms. These results show a fairly constant re­

turn loss of about 25 db throughout the frequency range. This seems to indicate th a t th e simulating network is a fairly close approxim ation to the artificial line so far as frequency is concerned and differs from it by a con­

sta n t multiplying factor which is of the order of 1.12. I t is therefore ap­

parent th a t whenever it is necessary only to sim ulate the impedance of this particular facility, this four-element network will provide a fairly adequate simulation. The analytical derivation of this network will be om itted.

method. This is the simulation of non-loaded cable of which the local plan t is largely composed in urban areas. A first approxim ation of the provides a three-elem ent network of the type discussed above which gives a retu rn of about 20 db in the 300 cycles to 3000 cycles range. T he graphical derivation of th e three-element netw ork is shown on Fig. 4 which also gives the impedance function for 22 ga. BSA non-loaded cable. This latte r func­

tion is virtually a straight line in the voice range whereas the network is the arc of a circle. Hence it would be impossible to obtain an appreciably closer approxim ation throughout the range with a three-element network.

However, th e addition of elements will improve the m atch as will be shown in example 3b.

The netw ork ju st derived can be expressed in term s of the 1000-cycle impedance and applied for any gauge of non-loaded cable as follows:

Ex a m p l e 3a— No n-l o a d e d Ex c h a n g e Ar e a Ca b l e

A nother case will be cited to show the application of th e graphical

analytical m ethod does not yield a useful netw ork b u t the graphical m ethod

R i = .42 K R t = 2.8 K X C2 = .9 K

(5-a) (5-b) (5-c) where K is th e m agnitude of the 1000-cycle impedance and

(5-d) T able I l i a gives a comparison of the netw ork and line impedances and the com puted retu rn loss for frequencies through the 200 to 3000 cycle range.

Ex a m p l e 3b— 1 9 -Ga u g e Qu a d d e d No n- Lo a d e d To l l Ca b l e

Two complete RC functions plus a resistance are required to give a good sim ulation for non-loaded toll cable when the sim ulation is carried through the voice and carrier frequency ranges. T he impedance function for 19 ga.

0 2 0 0 4 0 0 600 SOO 1000 1200 1400

R E S IS T A N C E

Fig. 4— G raphical design of tw o-term inal balancing netw ork for 22-ga. non-loaded exchange area cable.

Ta b l e I l l- a

22 Gauge Non-Quadded Non-Loaded B S A Exchange Area Cable

F r e q .

kc

L in e I m p e d a n c e N e tw o r k I m p e d a n c e C o m p u te d R e t u r n

L o s s — d b

R e c t . P o l a r R e c t . P o l a r

0 .2 915—j905 1287/44/7 1380—j725 1555/2778 15.0

0 .5 580—j565 8O8/ 44.2 705—j725 IO IO /45.8 19.1

1.0 415— j400 5 7 6 /4 4 .0 390— j460 6 0 3 /4 9 .7 2 5.2

2 .0 295—j280 4 0 7 /4 3 .5 285—§245 3 7 6 /4 0 .7 2 6 .6

3 .0 250—j220 3 3 3 /4 1 .3 260—j 165 3 0 8 /3 2 .4 21.2

toll cable is plotted on Fig. 5. T he m ethod followed in determ ining the elements is somewhat as follows: R i will be given by th e intercept of the function on the R axis and is 130 ohms. N ext look a t the low-frequency range determ ined by R3C3 and draw a semicircle which approxim ates th e given function in the range of 200-500 cycles. T he diam eter of this semicircle

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 289

determines R3 as 2100 ohms and R2 is then autom atically determ ined as the difference between the i?-intercept of the R3 semicircle and Ri , hence

= 4 2 0 — 130 = 290 ohms. To determ ine C3, choose the XC3 semicircle a t 500 cycles to intersect the R3 semicircle a t a point near the 500-cycle impedance of the cable impedance function, b u t make some allowance for the added negative reactance of the R2C2 generating function. The determ ination of C2 can be made in either of two ways. F irst an X Ci semicircle can be drawn a t 5000 cycles which intersects the i ?2 semicircle a t an impedance near the 5000-cycle impedance of the cable. T he impedance a t 1000 cycles can then be found graphically for R2C2 and R3C3 and added together to Ri . This

,00KC| 2 0

¡ Ê 5 *

\ /V 3

\

\

5kc\ ^ 11

i

1 /

-- 0— IM P E D A N C E F U N C T IO N FO R 19 GA QUADDED T O L L C A B L E , 110 F

IM P E D A N C E F U N C T IO N FO R N E T W O R K

R , R 2= 2 9 0 ~ R 3= 2 I 0 0 ~ l30<~ r W V —1 H W n

C2=.30MF C 3 =.725M F

Z | z 2 z 3

! K ^ * 3 I

S. 1 \

I K ? ^ \

- A

<5KC

V

\

\ v>

\ s . 2KC

.

.2KC

,.IK C - - - J_KÇ_ - ---

- 2 0 0

,.,-400

H O

<14

œ-600

- 800

200 4 0 0 800 1000

R E S IS T A N C E

1200

Fig. 5— G raphical design of tw o-term inal balancing netw ork for 19-ga. quadded non-loaded toll cable.

to ta l im pedance a t 1000 cycles should provide a good simulation of the 1000- cycle impedance of the cable. A second procedure for finding C2 would be to follow a somewhat reverse process: Determ ine the 1000 cycle Z for the R3C3 function and su b tract it from the 1000 cycle Z of th e cable. Choose C\

such th a t the intersection of the R2C2 semicircles is near the point deter­

m ined by the subtraction of R3C3 from the cable.

To avoid confusion of lines the construction circles have been om itted from this last drawing except to show the addition of the 1000-cycle im­

pedances. As m ay be seen this netw ork shown in Fig. 5 provides a rather good sim ulation throughout the frequency range above 200 cycles.

As pointed out earlier, if th e first guess is not a sufficiently good approxi­

m ation a second try can be m ade based on the evident shortcom ings of th e first try . In this case if a closer approxim ation is required up to 20 kc the next step m ight be to change C3 to .8 mf which would m ake the Z3 contribu­

tions above .1 kc somewhat less negative and would therefore raise th e n e t­

work locus. T hen changing R\ to 140 ohms would shift th e locus 10 ohms to th e right. T he resulting locus would be somewhat closer a t the upper frequencies b u t the change would not be necessary unless a rath er high degree of balance is required.

Ta b l e I II-b

19 Ga. Qtiadded Non-Loaded Toll Cable

F r e q . k c .

N e t w o r k I m p e d a n c e L in e I m p e d a n c e

C o m ­ p u t e d R e t u r n L o s s v s T h e o -

R e c t . P o l a r R e c t . P o l a r L in e -d b

.1 1515 —jl0 6 4 1852 / 35T9 1103—j 1093 1554 /4 4 .7 18.3

.2 867 — J893 1244 / 4 5 .8 783— J770 1097 / 4 4 .5 24.8

.5 488 — j493 693 /4 5 .3 501— j482 696 /4 3 .9 3 8.2

1.0 376 — j340 507 / 4 2 .1 361— J335 492 / 4 2 .8 3 6 .0

2 .0 271 — j254 3 7 1 .6 /4 3 .2 265— j229 350 / 4O.8 29.0

3 .0 217 — j203 2 9 7 .2 /4 3 .2 223— i l 80 287 / 38 .9 2 7 .8 5 .0 166 5— j 139 2 1 1 .3 /3 8 .0 187— j 131 228 / 3 5 .I 26.1 8 .0 145 5— j9 6 .8 1 7 2 .0 /3 2 .2 164— j94 189 / 29.8 25.8 10.0 140 0— j 7 4 .4 1 5 8 .0 /2 8 .0 155— J79.2 174 / 2 7 .I 26.5 16.0 134 — j4 7 .2 1 4 2 .1 /1 9 .5 145— j5 5 .0 155 / 2O.8 27.9 20.0 132 — j3 7 .9 137.9/ 1 6 .O 141— j4 5 .1 148 / H A 2 7.9 100 131 - j7 .3 1 3 1 .0 / 3 .2 130— j 14.0 131 / 6 .2 3 1.7

In general th e success of a tria l of th e graphical construction m ay be determ ined im m ediately by comparing about three frequencies of th e line and network.

Table I l l b gives the com puted netw ork impedance and th e line im ped­

ance. T he com puted retu rn loss is also given and equals or exceeds 25 db a t all frequencies above 200 cycles.

I t is apparent th a t th e resistance and condenser elements of th e generat­

ing functions are in descending order of m agnitude w ith increasing fre­

quency for the non-loaded cable the impedance locus of which is essentially a straight 45° line. As pointed out earlier, the series addition of such gener­

ating functions m ay be converted to a ladder stru ctu re5, whose sections will have a tapered characteristic rath er th an repetitive.

5 Appendix D of T ransm ission C ircuits by K . S. Jo hnson.

T W O - T E R M I N A L B A L A N C I N G N E T W O R K S 291

Re t u r n Lo ss

W hen designing such networks for balancing purposes, it has been found convenient to plot the function on a sheet such as Fig. 6 which divides the right half of the complex plane into circular regions such th a t all points on or within the boundary of a given region have a return loss against the netw ork 1 + jO equal to or greater th an th a t corresponding to the boundary.

These circles are determ ined by the return loss voltage ratio k and the ratio

R E S I S T A N C E

Fig. 6— Curves of co n stan t retu rn loss for the netw ork 1 ZO = 1 + / 0

of the line and network impedances. They m ay be computed from the equation

(1 + L / N ) / { 1 - L / N ) = k (6) By plotting the line and sim ulating netw ork loci on such a sheet it is generally possible to observe visually whether or not a given network meets the specified return loss requirem ent. If visual accuracy is not adequate,

it is always possible to m easure off N and L and the angle between them , spot th e complex ratio L / N on th e complex plane and read im m ediately the approxim ate re tu rn loss.

Co n c l u s i o n

The examples of the foregoing discussion have been confined to th e fourth q u ad ran t. I t was shown th a t by graphical m eans a num ber of parallel resistance-condenser functions could be determ ined which when added together would yield a close approxim ation to the given function. In the m ost general case these functions would involve th e generating function of R, L and C in parallel, th e locus of which is a circle having impedance + 7O a t zero frequency and —jO a t infinite frequency, and crossing the axis of reals a t R and th e frequency a t which L and C are anti-resonant. A case which has been found useful in sim ulating such things as telephone sets and other inductive elements is th e parallel com bination of R and L which, of course, is th e special case for C — 0 and occurs in th e first quadrant.

T he foregoing has been discussed w ith th e tho u g h t th a t it m ay be useful where there is lim ited time and where the required degree of sim ulation is consistent w ith a graphical m ethod. A t some future tim e it m ay be possible to pursue the problem further and devise the analytic counterpart to the somewhat heuristic graphical method.

C H A PTER III

The Use of X-Rays for Determining the Orientation of Quartz Crystals

By W. L. B O N D And E. J. AR M STR O N G

H IS paper is one of a series by the C rystal Research Group on the m anufacture of quartz oscillator plates. C ertain sections of it which are n o t original, b u t rath er adaptations of text book m aterial to the present problem, are included for purposes of completeness and for the convenience of those readers whose knowledge of the crystallographic literature may be limited.

3 .1 Pr o d u c t io n o f X - Ra y s f o r Qu a r t z Cr y s t a l X - Ra y Wo r k

X -rays are produced when electrons strike a m etal target a t high velocity.

The wave-length of X -rays given off from an X -ray tube varies from the longest which can pass through the X -ray tube window to the shortest th a t can be produced from the given target by the applied peak voltage. By analogy to the visible spectrum this is referred to as “ white” radiation.

For each different m etal, however, there are characteristic radiations of certain wave-lengths whose intensity m arkedly exceeds those of other wave­

lengths (Fig. 3.1). The strongest of these characteristic radiations is known as the K ai, the next strongest (generally half as strong and of slightly longer wave-length) as X a 2 and the third strongest (shorter in wave-length than K a i) is K(3. T he higher the atomic num ber of the target, the shorter will be the wave-length of the characteristic radiation. Therefore higher volt­

ages will be required to excite the characteristic radiation from the heavier m etals. (The minimum wave-length of X -rays th a t can be excited by any

1.234 X 10-4

given voltage is given by the equation Amin. = y where I is expressed in volts and A m i n . in Angstrom units).

H igher voltages also raise the intensity of the white radiation and, a t any given voltage, the white radiation produced from a heavy m etal target is more intense th an th a t produced from a lighter m etal target (see Figure 3.1). W hen “ w hite” radiation is desired, as in Laue photography, heavy m etal targets, such as tungsten, are used: when “monochromatic” radiation is desired, as in crystal goniometry, the lighter m etal targets, such as copper, are used because, w ith a lighter m etal target (wave-length of char­

acteristic radiation long) the voltage, and therefore the intensity of the 293

white radiation, cannot be raised very high before exciting the characteristic radiation whereas, w ith a heavy m etal ta rg et (wave-length of characteristic radiation short) considerable intensity of white radiation can be produced w ithout exciting the characteristic radiation. This is illustrated in Fig. 3.1 which shows th a t a p o tential of 35,000 volts is high enough to excite the K group radiation from m olybdenum , b u t n o t high enough to excite the shorter wave-length K radiation from tungsten which, further, gives more intense w hite radiation a t this voltage. E ven higher voltages, resulting in

Fig. 3.1—V ariation of intensity w ith w ave-length of X -ray s from tungsten and m olybdenum targets a t 35,000 volts

more intense white radiation, could be used w ith tungsten w ithout exciting the characteristic radiation (X = .209).

Figure 3.2 shows the I - \ curve (estim ated) for copper, the ta rg e t m etal commonly used in quartz X -ray work, which has a small atom ic num ber and can therefore be used as a source of “ m onochrom atic” X -rays w ith m oderate voltages. (A further advantage of copper for q uartz work is pointed out a t the end of this section).

T he K a i and K a i wave-lengths are so close together th a t for m ost uses of “m onochrom atic” radiation no a tte m p t is m ade to elim inate the K a 2 rad ia tio n .' The X/3 radiation, however, gives a distinct intensity peak of shorter wave-length which m ust be reduced as m uch as possible by use of a m etal filter having a high absorption coefficient for the Kf.3 radiation of